An oil company owns two refineries. The daily production limits and operating costs for each refinery are given in the table below. An order is received for 1540 barrels of high-grade oil, 1650 barrels of medium-grade oil, and 2860 barrels of low-grade oil. How many days should each refinery operate so that the order can be filled at the least cost?
Refinery 1 Refinery 2
High-grade oil (barrels) 110 220
Medium-grade oil (barrels) 220 110
Low-grade oil (barrels) 330 220
Operating cost (barrels) 14,000 8,000
X = _________
Y = _________
Constraints:
Objective Function: __________
Least Cost: ___________
To find the number of days each refinery should operate to fill the order at the least cost, we can use linear programming.
Let's consider X to be the number of days refinery 1 operates and Y to be the number of days refinery 2 operates.
We need to establish the constraints and the objective function for this problem.
Constraints:
1. High-grade oil constraint: 110X + 220Y >= 1540
The total production of high-grade oil from both refineries should be greater than or equal to 1540 barrels.
2. Medium-grade oil constraint: 220X + 110Y >= 1650
The total production of medium-grade oil from both refineries should be greater than or equal to 1650 barrels.
3. Low-grade oil constraint: 330X + 220Y >= 2860
The total production of low-grade oil from both refineries should be greater than or equal to 2860 barrels.
4. Non-negativity constraint: X, Y >= 0
The number of days each refinery operates cannot be negative.
Objective Function:
We need to minimize the operating cost. The operating cost for refinery 1 is $14,000 per day, and the operating cost for refinery 2 is $8,000 per day. Therefore, the total operating cost can be calculated as:
Total Cost = 14,000X + 8,000Y
Now, the problem can be formulated as follows:
Minimize: Total Cost = 14,000X + 8,000Y
Subject to:
110X + 220Y >= 1540
220X + 110Y >= 1650
330X + 220Y >= 2860
X, Y >= 0
To find the least cost, we need to solve this linear programming problem using a graphing method or a linear programming solver. The solution will provide the values for X and Y, representing the number of days each refinery should operate.